And you did not show that there must be a, b such that f(a,b)= (0,2), you simply asserted it! If f(a,b)= (0, 2), HOW would you find a and b? Exactly what must they be?

OK ya, got it. There cannot be any values of a&b which give the image as (0,2). [Since if x=0 then x*y=0 always). So the fuction is not surjective. There does not exist any value in the domain which is giving this particular value in the image.